Asymptotic Distributions and Berry-Esseen Bounds for Sums of Record Values

Qi-Man Shao (University of Oregon and National University of Singapore)
Chun Su (University of Science an Technology of China)
Gang Wei (Hong Kong Baptist University)

Abstract


Let $\{U_n, n \geq 1\}$ be independent uniformly distributed random variables, and $\{Y_n, n \geq 1\}$ be independent and identically distributed non-negative random variables with finite third moments. Denote $S_n = \sum_{i=1}^n Y_i$ and assume that $ (U_1, \cdots, U_n)$ and $S_{n+1}$ are independent for every fixed $n$. In this paper we obtain Berry-Esseen bounds for $\sum_{i=1}^n \psi(U_i S_{n+1})$, where $\psi$ is a non-negative function. As an application, we give Berry-Esseen bounds and asymptotic distributions for sums of record values.

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Pages: 544-559

Publication Date: June 25, 2004

DOI: 10.1214/EJP.v9-210

References

  1. Arnold, B.C. and Villasenor, J.A. (1998). The asymptotic distributions of sums of records, Extremes 1, 351-363. Math. Review 02a:60025
  2. Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation . Cambridge University Press, Cambridge.
  3. Chen, L.H.Y. and Shao, Q.M. (2003). Uniform and non-uniform bounds in normal approximation for nonlinear Statistics. Preprint.
  4. de Haan, L and Resnick, S.I. (1973). Almost sure limit points of record values. J. Appl. Probab. 10, 528--542. Math. Review 0372969 (51 #9171)
  5. Embrechts, P., Kl"{u}ppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  6. Hu, Z.S., Su, C. and Wang, D.C. (2002). The asymptotic distributions of sums of record values for distributions with lognormal-type tails. Sci. China Ser. A 45 , 1557--1566. Math. Review 04a:60054
  7. Mikosch, T. and Nagaev, A.V. (1998). Large Deviations of heavy-tailed sums with applications in insurance. Extremes 1, 81-110. Math. Review 99i:60057
  8. Petrov, V.V. (1995). Limit Theorems of Probability Theory, Sequences of Independent Random Variables . Clarendon Press, Oxford.
  9. Resnick, S.I. (1973). Limit laws for record values. Stoch. Process. Appl. 1, 67-82. Math. Review MR0362454 (50 #14895)
  10. Su, C. and Hu, Z.S. (2002). The asymptotic distributions of sums of record values for distributions with regularly varying tails. J. Math. Sci. (New York) 111, 3888--3894. Math. Review 04a:60102
  11. Tata, M.N. (1969). On outstanding values in a sequence of random variables. Z. Wahrsch. Verw. Gebiete 12, 1969 9--20. Math. Review 0247655
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